The Stacks project

Lemma 81.10.4. In Situation 81.2.1 let $X, Y, Z/B$ be good. Let $f : X \to Y$ and $g : Y \to Z$ be flat morphisms of relative dimensions $r$ and $s$ over $B$. Then $g \circ f$ is flat of relative dimension $r + s$ and

\[ f^* \circ g^* = (g \circ f)^* \]

as maps $Z_ k(Z) \to Z_{k + r + s}(X)$.

Proof. The composition is flat of relative dimension $r + s$ by Morphisms of Spaces, Lemmas 66.34.2 and 66.30.3. Suppose that

  1. $A \subset Z$ is a closed integral subspace of $\delta $-dimension $k$,

  2. $A' \subset Y$ is a closed integral subspace of $\delta $-dimension $k + s$ with $A' \subset g^{-1}(A)$, and

  3. $A'' \subset Y$ is a closed integral subspace of $\delta $-dimension $k + s + r$ with $A'' \subset f^{-1}(W')$.

We have to show that the coefficient $n$ of $[A'']$ in $(g \circ f)^*[A]$ agrees with the coefficient $m$ of $[A'']$ in $f^*(g^*[A])$. We may choose a commutative diagram

\[ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \ar[r] & W \ar[d] \\ X \ar[r] & Y \ar[r] & Z } \]

where $U, V, W$ are schemes, the vertical arrows are ├ętale, and there exist points $u \in U$, $v \in V$, $w \in W$ such that $u \mapsto v \mapsto w$ and such that $u, v, w$ map to the generic points of $A'', A', A$. (Details omitted.) Then we have flat local ring homorphisms $\mathcal{O}_{W, w} \to \mathcal{O}_{V, v}$, $\mathcal{O}_{V, v} \to \mathcal{O}_{U, u}$, and repeatedly using Lemma 81.4.1 we find

\[ n = \text{length}_{\mathcal{O}_{U, u}}( \mathcal{O}_{U, u}/\mathfrak m_ w\mathcal{O}_{U, u}) \]

and

\[ m = \text{length}_{\mathcal{O}_{V, v}}( \mathcal{O}_{V, v}/\mathfrak m_ w\mathcal{O}_{V, v}) \text{length}_{\mathcal{O}_{U, u}}( \mathcal{O}_{U, u}/\mathfrak m_ v\mathcal{O}_{U, u}) \]

Hence the equality follows from Algebra, Lemma 10.52.14. $\square$


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